I definitely would not lean on Justin's comment, which is open to interpretation. The way I read it, the second person reference in mathein dunasthe (1 Apology 34.2) is to the Emperor, his son, Lucius the Philosopher, and the Roman Senate (cf. the address in 1.1), i.e. high-level government officials whom Justin believed had access to official administrative records in Rome (cf. Tertullian, Adversus Marcionem, 4.7). Justin does not say that the records were public, or that he had seen them, or that he had independent verification of the enrollment of Jesus' family in the census records. He already had it on good authority (the authority of Luke, then recognized as Scripture) that Jesus' family was enrolled in that census, and he would not have doubted this information. His statement is not evidence per se that he had any direct information from the census itself, and it is a bit of a false dilemma to assume that Justin either was making a false statement or had direct knowledge of what was in the official archives.
The biggest obstacle imho to reading Luke as implying a different census than the one in AD 6 is the laconic reference to "the census" in Acts 5:37. Since Luke-Acts originally formed a single work, a reader of Acts would naturally find the logical antecedent of this census in Luke 2:1-3 which otherwise is the only census mentioned by the author. This passage in Acts states that the revolt of "Judas the Galilean" occurred "in the days of the census". In fact, Josephus mentions this revolt by Judas the Galilean and he says that it occurred as a reaction to the census of Quirinius after the replacement of Herod Archelaus with Coponius (Antiquities 18.1-6). That was the census of AD 6. So both Josephus and Luke-Acts associate Quirinius and Judas the Galilean with a census and if the author of Luke-Acts meant two different censuses, it is remarkable that he did not distinguish the two (especially since this "later" census would have ALSO been a census under Quirinius) and that this failure to do so just so happened to produce the same associations between Quirinius, the census, and Judas the Galilean found in Josephus. And if we grant the weight of the evidence that suggests that the author of Luke-Acts was dependent on Josephus, then he would have definitely meant the census of AD 6.
Other than that, there is the fact that the census in Luke 2:1-3 is presented as resulting from a decree by Augustus Caesar, indicating that Judea was then subject to direct Roman taxation. According to Josephus, it was the end of Herodian family rule (during which Judea was a client kingdom, not a Roman province) that brought this circumstance about, and he presents the census of AD 6 as the first systematic official Roman census of its kind -- representing a new assertion of Roman power in the province. This also accounts for the description of the census as the "first census" in Luke. The revolt of Judas the Galilean reacted to this change of affairs, which he viewed as nothing other than "an introduction to slavery" (Antiquities 18.1-6, Bellum Judaicum 2.433). If Luke presented the census as a local administrative affair, internal to the kingdom, that would be one thing, but making the census an imperial effort suggests that Judea at the time was subject to systematic imperial taxation, and that fits the famous census of AD 6 better than some unknown census earlier when Judea was still a kingdom under Herod.
